| Issue |
ESAIM: M2AN
Volume 57, Number 6, November-December 2023
|
|
|---|---|---|
| Page(s) | 3403 - 3437 | |
| DOI | https://doi.org/10.1051/m2an/2023085 | |
| Published online | 29 November 2023 | |
A convergent finite volume scheme for the stochastic barotropic compressible Euler equations
Centre for Applicable Mathematics, Tata Institute of Fundamental Research, P.O. Box 6503, GKVK Post Office, Bangalore 560065, India
* Corresponding author: ujjwal@math.tifrbng.res.in; toujjwal@gmail.com
Received:
31
December
2021
Accepted:
18
October
2023
In this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure-valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong convergence of numerical solutions to the regular solution of the limit systems at least on the lifespan of the latter, thanks to the weak (measure-valued)–strong uniqueness principle for the underlying system. To the best of our knowledge, this is the first attempt to prove the convergence of numerical approximations for the underlying system.
Mathematics Subject Classification: 35R60 / 60H15 / 65M08 / 76M12 / 76N10
Key words: Compressible fluids / Euler system / finite volume schemes / stochastic forcing / martingale solutions / dissipative measure-valued solution / weak–strong uniqueness entropy stable fluxes / convergence
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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